1.03 Checking reasonableness

So far we have been finding exact answers to arithmetical operations, and learning how to perform them quickly and accurately is an important skill. An equally important skill is being able to reasonably estimate answers - this is useful not only when speed is more important than accuracy, but also to check calculations to see if they are roughly on the right track.

Suppose you and two friends had to split more than \$10 evenly between you. One of your friends counts it out, and by the end you have \$1.50. You would be able to rightly complain that something went wrong - if all three of you ended up with only \$1.50 then the total amount that was split would have been less than \$5 to begin with.

Importantly, you don't need to know exactly how much you started with to check the reasonableness of this calculation. Instead you can let your friends know that they need to try again - and you would watch a lot more closely this time.

One of the most important tools we use when checking reasonableness is rounding the numbers to make the calculation easier. This produces an estimate - the more you round, the more inaccurate your estimate becomes. If you round the numbers up, then the estimate will be larger than the true answer, and if you round down, the estimate will be smaller.

The more you round, the less reasonable your calculation becomes - rounding each number in 6\times6 to the nearest 10 makes 10\times10, and 100 is not a good estimate for 6\times6=36.

Also, if you round some numbers up and some numbers down, you'll still produce an estimate, but you won't be able to easily tell if it's an overestimate or an underestimate. Rounding each number in \\ 21\times19 to the nearest 10 makes 20\times20=400, which is a good overestimate. Rounding each number in 18\times23 to the nearest 10 also makes 20\times20=400, which is a good underestimate.

To find a reasonable estimate for a calculation, try rounding the numbers to the nearest 10, the nearest 100, or some other place value. This will make the calculation easier and give you a rough idea of what the answer should be.

If you round all the numbers up, your new calculation will be an overestimate (larger than the original).

If you round all the numbers down, your new calculation will be an underestimate (smaller than the original).

Examples